A quick schematic road-map to these new geometric objects. The positroidron can be seen as a cellular structure on the nonnegative Grassmanian – the part of the real Grassmanian G(m,n) which corresponds to m by n matrices with all m by m minors non-negative. The cells in the cellular structure of the positroidron correspond to those matrices with the same (+,0) pattern for m by m minors. When m=1 we get a (spherical) simplex. When we project the positroidron using an n by k totally positive matrix we get for m=1 the cyclic polytope, and for general m the amplituhedron. When we project using general matrices we obtain general polytopes for m=1, and an interesting extension of polytopes proposed by Thomas Lam for general m.

Alex Postnikov’s recent lectures series in our Midrasha was an opportunity to understand slightly better some remarkable combinatorial objects that drew much attention recently. So here is my tentative partial understanding (based on Postnikov’s talks and some very useful explanation by Alex and by Lauren Williams) of it and some natural questions we can ask.

Cellular structures on the Grassmanian.

Schubert decomposition

The GrassmanianG(m,n), – the space of m-dimensional subspaces on an n dimensionsl space, can be seen as equivalence classes of m by n matrices under the action of GL(m). (So far, we can assume that the underlying field is arbitrary.)

The parts of the Schubert decomposition of G(m,n) correspond to all matrices with a given m-subset S with A as the lexicographically first non zero m by m minor.

Over the reals or the complex numbers this give an important cellular structure. The parts are indeed topologically open cells. Their closures are in general not homeomorphic to closed cells and can be quite (excitingly) complicated. The intersection of two cells can also be rather (excitingly) complicated.

The matroid decomposition

Given an n by m matrix we can regard its corresponding (representable) matroid. The bases of the matroid are those m-subsets S of [n] such that the corresponding m by m minor is non-zero. Two matrices belong to the same part in the matroid decomposition of G(m,n) if their corresponding matroids are the same. The matroid decomposition is the common refinement of the Schubert decomposition for all n! permutations of the n columns.

Parts in the matroid decomposition are realization spaces of matroids and they are very complicated and a whole area of universality theorems (with pioneering contributions by Nicolai Mnev) is devoted to that.

Postnikov decomposition: The cyclic refinement of the Schubert decomositions

Now, let us consider the common refinement of all Schubert decomposition with respect to the n cyclic permutation on [n]. It turns out that this is an important decomposition of the Grassmanian and we will call it the Postnikov decomposition.

The nonnegative Grassmanian over the reals

From now on we will work over the reals. Given an n by m matrix we can regard its corresponding oriented matroid as a map from m-subsets S of [n] to {+.-,0} which assigns to S the sign of the corresponding m by m minor. The possitive Grassmanian in G(m,n) consists of those matrices where all m by m minors are positive. The nonnegative Grassmanian consist of all matrices where all m by m minors are nonnegative. A positroid is a matroid which correspond to a matrix in the nonnegative Grassmanian. For the nonnegative Grassmanian, the Postnikov decomposition coincide with the matroid decomposition and they give us a remarkable cellular objects that we call the positroidron. (The common name is “the (positroid) cell decomposition of the positive (or non-negative) Grassmannian.”)

The triangle: m=1, n=2

When m=1 the positroidron is simply a simplex, or more precisely a certain spherical simplex.

An example m=2, n=4.

The postroidron is a four-dimensional CW complex that can be described as follows. The 2-skeleton is the 2 skeleton of the octahedron. First you add two of the three squares in the equator. (The symmetry breaking allowing to choose these two comes from the preferred cyclic ordering.) Then for each square you add the two pyramids on both its sides. (These are curved pyramids so they do not overlap.) Finally there is an additional four-dimensional face: the positive Grassmantan is a solid body whose boundary is the union of these four pyramids.

The combinatorial structure of the positroidron was explored by Alex Postnikov, and further combinatorial descriptions by several authors followed. It can be expressed in terms of certain planar graphs and it is closely related to certain permutation statistics.

The Amplituhedron and Lamtopes

The cyclic polytope and general polytopes.

Every convex polytope is a projection of the simplex. When we project using an n by k totally positive matrix we obtain the cyclic polytopes (A matrix is totally positive if all its minors are positive. Here it is enough to consider k by k minors.) Cyclic polytopes were discovered by Caratheodory at the early 20th century, and later rediscovered by David Gale who determined their combinatorial structure.

The amplituhedron and Lamtopes

The amplituhedron is a simple extension of the cyclic polytope for m>1. It is simply the projection of the positroidron under a totally positive matrix. This is a generalization of the cyclic polytope.

Thomas Lam (according to Alex Postnikov) proposed to consider not just the Amplituhedron but arbitrary projections of the positroidron.

The amplituhedron and physics

Computations of scattering amplitudes using the positive Grassmanians offers substantial simplification compared to “naive” Feynman diagrams computations. The computations are based on summing up certain volumes on cells of the positroidron, and the sums extend over cells which correspond to faces of the amplituhedron. This suggests that the cells of the amplituhedron have some conceptual significance. (It may also allow further computing improvement in the future.) Nima Arkani-Hamed and Jaroslav Trnka regard the amplituhedron as a key geometric object for understanding the emergence of locality and unitarity in quantum physics (or something beyond quantum physics).

Representability

The questions if a matroid is representable over a field, or if an oriented matroid is representable over the reals are very basic. A 1987 conjecture by da Silva asserts that (abstract) Positroids are representable and this was proved by Federico Ardila, Rincón, and Williams. The abstract analog of the Grassmanian (based on abstract oriented matroids rather than just representable ones) is called McPhersonian.

The Bruhat order

Important examples of regular CW deomposition of spheres come from intervals in the Bruhat order of Coxeter groups. It will be interesting to know if such intervals also appear (as are) as intervals in the geometric objects considered here. (Definitely Bruhat intervals of the form [e,w] (where w is a Grassmannian permutation) appear as intervals in the poset of cells in the positive Grassmannian. This is more or less immediate from the description of the poset of cells of the positive Grassmannian in terms of pairs of permutations.)

Some connections between the positroidron and the Bruhat order are known and studied in Lauren Williams’ paper Shelling totally nonnegative flag varieties . The appendix to the paper describes some early combinatorial representations of the positroidron.

Open questions

A basic questions:

1) Is the positroidron a regular CW complex? As far as I know it was proved already by Postnikov that the open cells are homeomorphic to an open ball and Konni Rietsch and Lauren Williams proved that the closed cells are contractible, with boundaries homotopy-equivalent to spheres — in Discrete Morse theory for totally non-negative flag varieties. It is conjectured that closures of open cells are homehomorphic to close balls.

The results and techniques in Hersh’s paper on regular CW complexes in total positivity are closely related.

Further question

2) Are the amplituhedron and Lamtopes are regular CW complexes? In fact, is it the case that their open cells are homehomorphic to balls?

3) (Perhaps well understood) how does intervals in the Bruhat order of Coxeter groups fits into the picture.

4) Describe the cellular structure of the amplituhedron.

5) Are Lamtopes closed under taking intervals and duality. If not, what would be an interesting class of objects including Lamtopes which are closed to taking intervals and to duality.

A reminder: miracles of cyclic polytopes

a) Cyclic polytopes in d dimensions are neighborly. When you regard it as a projection of the simples, every face of the simplex of dimension [d/2]-1 is mapped to a face of the cyclic polytope.

b) Cyclic polytopes are extremal. The upper bound theorem asserts that the maximum of k-faces for a d-polytopes with n vertices is attained by the cyclic polytope.

c) The vertices of the cyclic polytope represent a remarkable oriented matroid or “order-type.” It is universal in the following sense: For a set of points in in “cyclic position” every subset is also in cyclic position. On the other hand, for every n and d there is f(n,d) such that every set of f(n,d) points in contains a subset of n points in cyclic position. In the plane this is the famous Erdos Szekeres theorem. In high dimension this is now well understood by works of Jirka Matousek and a few coauthors.

Further question motivated by the cyclic polytope and related theory

6) Is the amplituhedron extremal among Lamtopes in terms of face numbers and other combinatorial parameters? (The upper bound theorem asserts that cyclic polytopes are extremal among polytopes.) Does the amplituhedron exhibit neighborliness? tightness?

7) Do the notions of g-polynomial [and Kazhdan-Lusztig polynomial] extend to lamtopes?

8) (Proposed by Eric Katz) Do the notions of secondary polytopes/fiber polytopes extend to lamtopes.

9) Is the amplituhedron (in some sense) “neighborly?”

10) Are the vertices of the amplituhedron (thought of as representing a sequence of k-spaces in some Euclidean space) universal in some sense?

Updates: Some additional things.

1) (Feb. 22) Hersh’s work: The paper “Regular cell complexes in total positivity” shows that certain spaces of totally nonnnegative, real matrices, stratified according to which minors are positive and which are 0, are regular CW complexes homeomorphic to closed balls having the closed intervals in Bruhat order as their posets of closure relations. These regular CW balls arise as links of cells in the double Bruhat stratification of the totally nonnegative part of the flag variety, (as can be seen from the Marsh-Rietsch parametrization of the totally nonnegative part of the flag variety). So this also gives some further evidence for the conjecture about the nonnegative part of the flag variety being a closed ball. A main example of the stratified spaces Hersh studied was the totally nonnegative, real, upper triangular matrices with 1’s on the diagonal and entries just above the diagonal summing to a fixed positive constant, stratified according to which minors are positive and which are 0. (To me this shows that the actual picture is more general than described in this post.)

2) (Feb 22) Algebraic shifting: In my early works I studied stratifications of Grassmanians G(V,m) where V itself is a kth exterior power of an n-dimensional vector space. Algebraic shifting is essentially the study of the Schubert decomposition with respect to the lexicographic ordering of base elements of the exterior algebra (which are indexed by k-subsets of [n].). In general, it is interesting to study stratifications of the Grassmanian when the vector space we start with has additional structure.

Trivia question

When you search pictures in Google for “Bruhat order” you get this picture.

The proof is based on two major ingredients. The first is a recent major theory by Issak Mabillard and Uli Wagner which extends fundamental theorems from classical obstruction theory for embeddability to an obstruction theory for r-fold intersection of disjoint faces in maps from simplicial complexes to Euclidean spaces. An extended abstract of this work is Eliminating Tverberg points, I. An analogue of the Whitney trick. The second is a result by Murad Özaydin’s from his 1987 paper Equivariant maps for the symmetric group, which showed that for the non prime-power case the topological obstruction vanishes.

It was commonly believed that the topological Tverberg conjecture is correct. However, one of the motivations of Mabillard and Wagner for studying elimination of higher order intersection was that this may lead to counterexamples via Özaydin result. Isaak and Uli came close but there was a crucial assumption of large codimension in their theory, which seemed to avoid applying the new theory to this case. It turned out that a simple combinatorial argument allows to overcome the codimension problem!

Florian’s combinatorial argument which allows to use Özaydin’s result in Mabillard-Wagner’s theory is a beautiful example of a powerful combinatorial method with other applications by Pavle Blagojević, Florian Frick and Günter Ziegler.

Both Uli and Florian talked about it here at Oberwolfach on Tuesday. I hope to share some more news items from Oberwolfach and from last week’s Midrasha in future posts.

Update 3 (January 30): The midrasha ended today. Update 2 (January 28): additional videos are linked; Update 1 (January 23): Today we end the first week of the school. David Streurer and Peter Keevash completed their series of lectures and Alex Postnikov started his series.

___

Today is the third day of our winter school. In this page I will gradually give links to to various lectures and background materials. I am going to update the page through the two weeks of the Midrasha. Here is the web page of the midrasha, and here is the program. I will also present the posters for those who want me to: simply take a picture (or more than one) of the poster and send me. And also – links to additional materials, pictures, or anything else: just email me, or add a comment to this post.

It has been a while since I devoted a post to quantum computation. Meanwhile, we had a cozy, almost private, easy-going, and very interesting discussion thread on my previous, March 2014 post (that featured my Simons Institute videotaped lectures (I,II).)

What can we learn from a failure of quantum computers?

Last week we had a workshop on “Quantum computing: achievable reality or unrealistic dream.” This was a joint venture of the American Physics Society and the Racah Institute of Physics here at HUJI, organized by Professor Miron Ya. Amusia, and it featured me and Nadav Katz as the main speakers. Here are the slides of my lecture: What can we learn from a failure of quantum computers.

On the positive side, Greg Kuperberg and I wrote a paper Contagious error sources would need time travel to prevent quantum computation showing that for a large class of correlated noise, (teleportation-based) quantum fault-tolerance works! Greg and I have had a decade-long email discussion (over 2000 emails) regarding quantum computers, and this work grew from our 2009 discussion (about my “smoothed Lindblad evolution” model), and heavily relies on ideas of Manny Knill.

Nadav Katz: Quantum information science – the state of the art

Some years ago, two brilliant experimentalists, Hagai Eisenberg and Nadav Katz, joined the already strong, mainly theoretical, quantum information group here at HUJI. Nadav Katz gave the second lecture in the workshop, and here are the slides of Nadav’s lecture: Quantum information science – the state of the art.

Experimental progress toward stable encoded qubits

Also very much on the positive side, Nadav mentioned a remarkable recent progress by the Martini’s group showing certain encoded states based on 9 physical qubits which are order-of-magnitude (factor 8.4, to be precise,) more stable than the “raw” qubits used for creating them !!

Update: Further comments on a Shtetl-optimized post (especially a comment by Graeme Smith,) help to place the new achievement of the Martinis group within the seven smilestones toward quantum computers from a 2012 Science paper by Schoelkopf and Devoret, originated by David DiVincenzo’s 2000 paper “The physical implementation of quantum computation“. (You can watch these milestone here also .)

The new achievement of having a very robust realization of certain encoded states can be seen as achieving the 3.5 milestone. The difference between the 3.5th milestone and the 4th milestone plays a central role in the seventh post of my 2012-debate with Aram Harrow in connection with a conjecture I made in the first post (“Conjecture 1″). Aram made the point that classical error-correction can lead to very stable encoded qubits in certain states (which is essentially the 3.5 milestone). I gave a formal description of the conjecture, which essentially asserts that the 4th milestone, namely insisting that encoded qubits allows arbitrary superpositions, cannot be reached. As I said many times (see, for example, the discussion in my 2012 Simons Institute videotaped lecture 2), a convincing demonstration of the 4th milestone, namely implementation of quantum error-correction with encoded qubits which are substantially more stable than the raw qubits (and allow arbitrary superposition for the encoded qubit) will disprove my conjectures. Such stable encoded qubits are expected from implementations of distance-5 surface code. So we are 0.5 milestones away :)

I will be impressed to see even a realization of distance-3 (or distance-5) surface code that will give good quality encoded qubits, even if the encoded qubits will have a quality which is somewhat worse than that of the raw qubits used for the encoding. These experiments, including those that were already carried out, also give various other opportunities to test my conjectures.

Peter Shor’s challenge #1 and my predictions from the failure of quantum computation

My lecture on predictions from the failure of QC is based on two lengthy recent comments (first, second) regarding predictions from the failure of quantum computers. On April 2014, Peter Shor challenged me with the following:

So if “quantum computers cannot work” is the premise of your theory, does your theory lead to any concrete predictions besides “quantum computers cannot work”? If it doesn’t, then I can’t help thinking that there really isn’t much substance to it.

While the discussion continued, it was not until November that I gave a detailed reply – a list of around 18 concrete predictions, along with other connections, possible applications, further vaguer predictions and speculations. (The comments have a much wider scope than the lecture.)

Following my longish prediction comments, Peter challenged me with yet another great question.

We already know that the standard formulation of quantum physics appears to be computationally difficult (I am talking about the formulation without smoothed Lindblad evolution).

What you are saying is that introduced noise will make the system behave differently from its current predictions and make it easier to simulate.

Right now, particle physics experiments are simulated using lattice QFT, this simulation is computationally very intensive, and simulating particle physics is believed to high computational complexity. If we extrapolate your prediction, you predict that particle physics experiments will actually behave differently from the standard predictions, and in a way that is easier to simulate.

Is this correct, or is particle physics for some reason immune to your hypothesis?

The very short answer is that particle physics is not immune to my hypothesis!

This is an extremely interesting topic that a few of us here at HUJI have been thinking about together in the last year or so. What are the best examples for “quantum supremacy” in the context of many-body quantum systems (be it from condensed matter physics, chemistry, nuclear or atomic physics, or high energy physics), and how can we test and challenge (even on a heuristic level) the quantum-supremacy claim. (This is a difficult conceptual issue and we also looked briefly on connections with chaotic behavior in classical systems.) We discussed quite a few potential candidates like energy levels and other features of heavy atoms, highly accurate atomic clocks, and more, and right now, the best candidate we have is QED computations for energy levels of the Hydrogen atom.

I am not sure what is stronger: the fact that, in quantum physics, computations are lagging behind experiments, sometimes by several order of magnitudes — as an argument in favour of QC as a realistic possibility, or the (likely) fact that QC will enable to perform quantum physics computations, hundreds of orders of magnitude more accurately than experiments –as an argument against QC as a realistic possibility.

Michel and I, and Scott

Miron’s initial plan was to invite Michel Dyakonov, a prominent theoretical physicist with strong skeptical views regarding QC. While we are both skeptics regarding QC, Michel’s views are different from mine, see the pictures at the top. (Here is a link to a presentation of Michel from 2009 explaining his thoughts on the matter.) Michel who visited HUJI last winter does not expect failure of QC to lead to important physics predictions, but he prefers to draw some strong sociological insights. This winter, Scott Aaronson, who also has strong views on the matter of QC, was in town and it was fun to spend some time with him and chat face-to-face.

I remember a lengthy discussion on “polymath4″ with many contributions from both Noam Nisan and me, and then how much better it was to actually meet Noam, in person, in the cafeteria one day. (This gap is even larger when it comes to debates and disagreements.) When I mentioned this to Terry Tao he told me that in Internet discussions he can actually “visualize” the other participants so it is almost like talking with them in person.

Updates: 1) a related Shtetl-optimized post describes the Martini’s group experimental breakthrough as well as a theoretical one by Ed Farhi, Jeffrey Goldstone, and Sam Gutmann about a quantum algorithm for approximation of an NP-hard optimization problem.

2) Michel’s picture at the top of this post was designed as an illustration to Michel’s 2001 presentation “Quantum computing: a view from the enemy camp” . The “enemy camp” is presented on the right side of the picture. The actual drawing was done by Michel’s son, 16 years old at that time. ( I remember that I when I read the paper I felt, or imagined, some “anti-mathematics” sentiment. When I asked Michel about it last year, he told me that his son is actually a mathematician! )

In the pictures you can see also Miron, Nadav, Guy and Scott, and the stormy sea at the Tel-Aviv old harbor as taken by Guy.

A videotaped lecture on BosonSampling

Update: A video of the debate

]]>https://gilkalai.wordpress.com/2015/01/10/quantum-computing-achievable-reality-or-unrealistic-dream/feed/7gilkalaiQC-michel-viewQC-gilviewmy_qc_worldpred1nadavtalkmiron's workshopws4ws1ws2ws3DSC04300PENTAX ImageALLmIMG_4693A Historical Picture Taken by Nimrod Megiddohttps://gilkalai.wordpress.com/2014/12/31/a-historical-picture-taken-by-nimrod-megiddo/
https://gilkalai.wordpress.com/2014/12/31/a-historical-picture-taken-by-nimrod-megiddo/#commentsWed, 31 Dec 2014 18:13:46 +0000http://gilkalai.wordpress.com/?p=12375Continue reading →]]>Last week I took a bus from Tel Aviv to Jerusalem and I saw (from behind) a person that I immediately recognized. It was Nimrod Megiddo, from IBM Almaden, one of the very first to relate game theory with complexity theory, one of the pioneers of computational geometry, and one of the leaders in optimization and linear programming, the guy who (with Ehud Kalai) was the first to invited me to an international conference, and a fresh Laureate of the John von Neumann theory Prize. I did not see Nimrod more than a year after our last coincidental meeting at the Berkeley Simons Institute, I called over to him and he was happy to see me as I was happy to see him, and we found a place together at the back of the bus and caught up on things.

]]>https://gilkalai.wordpress.com/2014/12/31/a-historical-picture-taken-by-nimrod-megiddo/feed/0gilkalaiMegiddoScott Triumphs* at the Shtetlhttps://gilkalai.wordpress.com/2014/12/31/scott-triumphs-at-the-shtetl/
https://gilkalai.wordpress.com/2014/12/31/scott-triumphs-at-the-shtetl/#commentsWed, 31 Dec 2014 08:58:47 +0000http://gilkalai.wordpress.com/?p=12341Continue reading →]]>Scott Aaronson wrote a new post on the Shtetl Optimized** reflecting on the previous thread (that I referred to in my post on Amy’s triumph), and on reactions to this thread. The highlight is a list of nine of Scott’s core beliefs. This is a remarkable document and I urge everybody to read it. Yes, Scott’s core beliefs come across as feminist! Let me quote one of them.

7. I believe that no one should be ashamed of inborn sexual desires: not straight men, not straight women, not gays, not lesbians, not even pedophiles (though in the last case, there might really be no moral solution other than a lifetime of unfulfilled longing). Indeed, I’ve always felt a special kinship with gays and lesbians, precisely because the sense of having to hide from the world, of being hissed at for a sexual makeup that you never chose, is one that I can relate to on a visceral level. This is one reason why I’ve staunchly supported gay marriage since adolescence, when it was still radical. It’s also why the tragedy of Alan Turing, of his court-ordered chemical castration and subsequent suicide, was one of the formative influences of my life.

!!

In the sacred tradition of arguing with Scott I raised some issues with #5 and 4# on Scott’s blog. Two of Scott’s points are on the subject of (young) people’s suffering by feeling unwanted, sexually invisible, or ashamed to express their desires.

I was pleased to see that those feminist matters that Scott and I disagree about, like the nature of prostitution, the role of feminist views in men’s (or nerdy men’s) suffering, and also Scott’s take on poverty, did not make it to Scott’s core beliefs.

Happy new year, everybody!

* The word triumph is used here (again) in a soft (non-macho) way characteristic to the successes of feminism. Voting rights for women did not exclude voting rights for men, and Scott’s triumph does not mean a defeat for any others; on the contrary.

** “Shtetl-optimized” is the name of Scott Aaronson’s blog.

]]>https://gilkalai.wordpress.com/2014/12/31/scott-triumphs-at-the-shtetl/feed/6gilkalaiAmy Triumphs* at the Shtetlhttps://gilkalai.wordpress.com/2014/12/27/amy-triumphs-at-the-shtetl/
https://gilkalai.wordpress.com/2014/12/27/amy-triumphs-at-the-shtetl/#commentsSat, 27 Dec 2014 16:31:40 +0000http://gilkalai.wordpress.com/?p=12322Continue reading →]]>It was not until the 144th comment by a participants named Amy on Scott’s Aaronson recent Shtetl-optimized** post devoted to a certain case of sexual harassment at M. I. T. that the discussion turned into something quite special. Amy’s great comment respectfully disagreeing with the original post and most of the 100+ earlier comments gave a wide while personal feminist perspective on women in STEM (STEM stands for science, technology, engineering, mathematics). This followed by a moving comment #171 by Scott describing a decade of suffering from his early teens. Scott, while largely sympathetic with the feminist cause, sees certain aspects of modern feminism as major contributors to his ordeal.

Then came a few hundred comments by quite a few participants on a large number of issues including romantic/sexual relations in universities, rape, prostitution, poverty, gaps between individuals’ morality and actions, and much more. Many of the comments argued with Amy and a few even attacked her. Some comments supported Amy and some proposed their own views. Many of the comments were good and thoughtful and many gave interesting food for thought. Some people described interesting personal matters. As both Scott and Amy left school early to study in the university, I also contributed my own personal story about it (and Scott even criticized my teenage approach to life! :) ). Amy, over 80+ thoughtful comments, responded in detail, and her (moderate) feminist attitude (as well as Amy herself) stood out as realistic, humane, and terribly smart.

* The word triumph is used here in a soft (non-macho) way characteristic to the successes of feminism. Voting rights for women did not exclude voting rights for men, and Amy’s triumph does not mean a defeat for any others; on the contrary.